That is an interesting build you have there. Not a bad comment as I have two home-brew hydros undergoing development at the moment.
Your thrust angle is a quite aggressive therefore will lift the rear regardless of the type of prop, however a lifting prop will increase the effect.
A 1/8" shaft is fine to work with for a wide variety of props. You can use 3/16"octuras with a sleeve of 7/32" K&S tubing, I do !!
I do not know what rpm you are developing on the bench, but I get close to 25,000 rpm and use a octura 534 on my 20" prototype hydro with a solid shaft similar to yours.
If you have wads of rpm on hand go for a small prop and work steadily upwards. Keep a diary of changes.
I run for 30 - 60 seconds on a prop then swap to monitor changes. Given the cells I use I get 2 or 3 swaps per charge, once they are off the peak then i just run the boat for a nother min or two before breaking out the CAT and bigger mono.
Summary. Small prop to begin with, I suggest a 534 ( plastic ) and go from there.